On Strongly $\Delta$-Clean Rings

TL;DR

This paper introduces and analyzes strongly $ riangle$-clean rings, a new class where each element decomposes into a commuting idempotent and an element from an extended Jacobson radical, revealing their structural properties and relationships with other ring classes.

Contribution

It defines strongly $ riangle$-clean rings, explores their properties, and establishes criteria for their presence in various algebraic constructions, advancing the understanding of ring decompositions.

Findings

All strongly $ riangle$-clean rings are strongly clean and $ riangle U$.

Under certain conditions, they refine the class of uniquely clean rings.

Criteria are provided for triangular matrix rings, skew versions, and group rings.

Abstract

This study explores in-depth the structure and properties of the so-called {\it strongly $\Delta$-clean rings}, that is a novel class of rings in which each ring element decomposes into a sum of a commuting idempotent and an element from the subset $\Delta(R)$. Here, $\Delta(R)$ stands for the extension of the Jacobson radical and is defined as the maximal subring of $J(R)$ invariant under the unit multiplication. We present a systematic framework for these rings by detailing their foundational characteristics and algebraic behavior under standard constructions, as well as we explore their key relationships with other well-established ring classes. Our findings demonstrate that all strongly $\Delta$-clean rings are inherently strongly clean and $\Delta U$, but under centrality constraints they refine the category of uniquely clean rings. Additionally, we derive criteria for the strong $\Delta$-clean property in triangular matrix rings, their skew analogs, trivial extensions, and group rings. The analysis reveals deep ties to boolean rings, local rings, and quasi-duo rings by offering new structural insights in their algebraic characterization.